Develop two-null hypothesis tests to ascertain if there are any statistical variances in the averages of bus route times and any statistical variance among the average performance of the bus drivers. Use the following information to conduct the test:
The three bus routes are the treatments.
The three bus drivers are the blocks.
The common element is driving time.
Route 1 Time
Route 2 Time
Route 3 Time
Driver 1
24 minutes
21 minutes
27 minutes
Driver 2
28 minutes
27 minutes
25 minutes
Driver 3
18 minutes
13 minutes
19 minutes
To conduct hypothesis tests to ascertain if there are any statistical variances in the averages of bus route times and among the average performance of the bus drivers, we can use Analysis of Variance (ANOVA) tests.
For the bus route times, we can use a one-way ANOVA test. The null hypothesis would be that there is no statistical variance in the averages of the bus route times. The alternative hypothesis would be that there is statistical variance in the averages of the bus route times.
To perform the test, follow these steps:
1. Calculate the mean for each bus route (Route 1, Route 2, Route 3).
2. Calculate the overall mean of all the bus route times.
3. Calculate the sum of squares for treatments (SST), which measures the variation between the means of the bus routes.
4. Calculate the sum of squares for error (SSE), which measures the variation within each bus route.
5. Calculate the degrees of freedom for treatments (dfT) and the degrees of freedom for error (dfE).
6. Calculate the mean square for treatments (MST = SST / dfT) and the mean square for error (MSE = SSE / dfE).
7. Calculate the F-statistic (F = MST / MSE).
8. Compare the obtained F-statistic with the critical F-value from an F-distribution table or use an appropriate statistical software.
9. If the obtained F-statistic is greater than the critical F-value, reject the null hypothesis and conclude that there is statistical variance among the averages of the bus route times.
For the average performance of the bus drivers, we can use a two-way ANOVA test. The null hypothesis would be that there is no statistical variance among the average performance of the bus drivers. The alternative hypothesis would be that there is statistical variance among the average performance of the bus drivers.
To perform the test, follow these steps:
1. Calculate the mean for each bus route (Route 1, Route 2, Route 3).
2. Calculate the mean for each bus driver (Driver 1, Driver 2, Driver 3).
3. Calculate the overall mean of all the bus route times.
4. Calculate the sum of squares for treatments (SST), which measures the variation between the means of the bus routes.
5. Calculate the sum of squares for blocks (SSB), which measures the variation between the means of the bus drivers.
6. Calculate the sum of squares for error (SSE), which measures the variation within each bus route and driver combination.
7. Calculate the degrees of freedom for treatments (dfT), the degrees of freedom for blocks (dfB), and the degrees of freedom for error (dfE).
8. Calculate the mean square for treatments (MST = SST / dfT), the mean square for blocks (MSB = SSB / dfB), and the mean square for error (MSE = SSE / dfE).
9. Calculate the F-statistic for treatments (F = MST / MSE) and the F-statistic for blocks (F = MSB / MSE).
10. Compare the obtained F-statistics with the critical F-values from an F-distribution table or use an appropriate statistical software.
11. If either of the obtained F-statistics is greater than the corresponding critical F-value, reject the null hypothesis and conclude that there is statistical variance among the average performance of the bus drivers.
Remember, interpreting the results of hypothesis tests requires considering the significance level, sample size, and effect sizes.